Quantum spin-½ models serve as basic paradigms for a wide variety of physical systems in quantum statistical mechanics and many-body physics, and complex quantum spin-½ models are among the most studied from the perspective of quantum phase transitions and topological order. See Vi Zhou, Kazushi Kanoda, and Tai-Kai Ng, “Quantum spin liquid states,” Rev. Mod. Phys. 89, 025003 (2017); John Michael Kosterlitz, “Nobel lecture: Topological defects and phase transitions,” Rev. Mod. Phys. 89, 040501 (2017); F. Duncan M. Haldane, “Nobel lecture: Topological quantum matter,” Rev. Mod. Phys. 89, 040502 (2017); Cristiano Nisoli, Roderich Moessner, and Peter Schiffer, “Colloquium: Artificial spin ice: Designing and imaging magnetic frustration” Rev. Mod. Phys. 85, 1473-1490 (2013).
In addition, since the spin-½ particle in a magnetic field is one of the simplest realizations of a qubit, many quantum information processing paradigms draw heavily on concepts which originated from, or are related to, quantum magnetism. For example, the formalism underlying quantum fault-tolerance in gate-model quantum computing is built on viewing decoherence in a Pauli basis, and indeed in most cases physically engineering measurements to project errors onto this basis. See, for example: I. L. Chuang, R. Laflamme, P. W. Shor and W. H. Zurek, Quantum computers, factoring and decoherence, Science, 270, pp. 1635-1637 (1995); D. P. DiVincenzo and P. W. Shor, Fault tolerant error correction with efficient quantum codes, Phys. Rev. Lett. 77, pp. 3260-3263 (1996); Barbara M. Terhal, “Quantum error correction for quantum memories,” Rev. Mod. Phys. 87, 307{346 (2015), and reference therein; see also Andrew W. Cross, David P. Divincenzo, and Barbara M. Terhal, “A comparative code study for quantum fault tolerance,” Quantum Info. Comput. 9, 541-572 (2009).
In fact, nearly all constructions underlying quantum computational and error-mitigation protocols in quantum information processing are viewed in the language of effective quantum spin models, including those used in quantum annealing, Hamiltonian, and holonomic computing, quantum simulation, and boson sampling. Arnab Das and Bikas K. Chakrabarti, “Colloquium: Quantum annealing and analog quantum computation,” Rev. Mod. Phys. 80, 1061-1081 (2008); Zhang Jiang and Eleanor G. Rieffel, “Non-commuting two-local Hamiltonians for quantum error suppression,” Quantum Information Processing 16, 89 (2017); Seth Lloyd and Barbara M Terhal, “Adiabatic and Hamiltonian computing on a 2d lattice with simple two-qubit interactions,” New Journal of Physics 18, 023042 (2016); Daniel Nagaj and Pawel Wocjan, “Hamiltonian quantum cellular automata in one dimension,” Phys. Rev. A 78, 032311 (2008); Tzu-Chieh Wei and John C. Liang, “Hamiltonian quantum computer in one dimension,” Phys. Rev. A 92, 062334 (2015); Milad Marvian and Daniel A. Lidar, “Error suppression for Hamiltonian quantum computing in Markovian environments,” Phys. Rev. A 95, 032302 (2017); Paolo Zanardi and Mario Rasetti, “Holonomic quantum computation,” Physics Letters A 264, 94 (1999); Jacob T. Seeley, Martin J. Richard, and Peter J. Love, “The Bravyi-Kitaev transformation for quantum computation of electronic structure,” Journal of Chemical Physics 137, 224109 (2012); I. M. Georgescu, S. Ashhab, and Franco Nori, “Quantum simulation,” Rev. Mod. Phys. 86, 153{185 (2014); P. J. J. O'Malley, et al., “Scalable quantum simulation of molecular energies,” Phys. Rev. X 6, 031007 (2016); Diego Gonzalez Olivares, Borja Peropadre, Alan Aspuru-Guzik, and Juan Jose Garda-Ripoll, “Quantum simulation with a boson sampling circuit,” Phys. Rev. A 94, 022319 (2016); and Borja Peropadre, Alan Aspuru-Guzik, and Juan Jose Garda-Ripoll, “Equivalence between spin Hamiltonians and boson sampling,” Phys. Rev. A 95, 032327 (2017).
It is notable, then, that no physical qubit system implemented to date can realize strong, engineerable, static, vector spin-½ interactions. Instead, existing techniques rely on pulsed implementation of effective Hamiltonians. For example, in the context of digital quantum simulation, discrete, sequential, non-commuting gate operations can be used to simulate evolution under complex spin Hamiltonians (using the Suzuki-Trotter expansion), where the error relative to the static Hamiltonian being simulated can be reduced by shortening the individual discrete pulses (relative to the energies being simulated). The circuits devised to implement quantum fault-tolerance operate in a similar manner (with the addition of projective measurement), effectively simulating damping into a desired set of logical code states which are themselves built out of physical spin-½-like states of the constituent physical qubits.
Although these pulsed methods are powerful and flexible, they tend to encounter problems when the spin models require very strong and/or complex interactions. Time-domain, Trotterized implementation of complex spin Hamiltonians places very strong restrictions on the effective energy scales of the Hamiltonian being simulated, which can become a major roadblock. The number of gate operations required to approximate each time step of the full Hamiltonian grows quickly as the complexity of the interactions grows, causing the overall simulated energy scale to decrease. Since in real implementations there will always be a hard, practical upper limit to the frequencies of pulsed gates that can be applied without exciting additional spurious degrees of freedom, this results in an effective dynamic range that will be reduced as the complexity and/or the size of the Hamiltonian increase. In addition, gate-based implementation necessarily implies that the physical system occupies Hilbert space far above the ground state of the physical qubits undergoing the operations, resulting in a sensitivity to dissipative decoherence processes which tend to take the system irreversibly out of the encoded solution space (this is the motivation for quantum error correction). By comparison, a static Hamiltonian implementation has an intrinsic “protection” from noise associated with being in or near the ground state, both against logical fluctuations within the encoded solution space and against transitions out of this space. For these reasons, a qubit implementation capable of supporting strong, static, vector spin interactions could be instrumental for several applications. However, no such implementation has yet been found.
Superconducting circuits are already among the most engineerable high-coherence quantum systems available, allowing a range of behavior and interactions to be constructed by design. This is perhaps best exemplified using flux qubits to emulate quantum spin models for quantum annealing applications. Programmable quantum transverse-field Ising models with more than 2000 physical spins have been realized by D-Wave Systems, Inc. of Burnaby, Canada. However, there are fundamental limitations of flux qubits which prevent them from being used to realize strong, fully-programmable vector spin interactions.
FIGS. 1A-1D illustrate the physics behind this. FIG. 1A schematically shows an effective, flux tunable Josephson junction 112 inside a flux qubit 114 realized using a DC SQUID 116. When the flux qubit loop 118 is biased with an external flux Φe=Φ0/2 (where Φ0=h2e is the superconducting fluxoid quantum), the two lowest-energy semiclassical states of the loop 118 are nearly degenerate due to the Meissner effect, having approximately equal and opposite supercurrents flowing, corresponding to either expulsion of the external flux from the loop 118 or pulling additional flux into it, such that it contains exactly zero or one fluxoid quantum, respectively. These two semiclassical states are associated with two local minima 122 and 124 in an inductive potential 126 shown in FIG. 1B. This potential 126 is experienced by a fictitious particle, whose “position” corresponds to the fluxoid contained in the loop inductor. As shown in FIG. 1B, the two eigenstates of Z correspond to opposite persistent currents in the qubit loop, connected by a tunneling barrier 128 whose height is controlled by the flux through the DC SQUID 116, and they can be used to emulate the two eigenstates of Z for an effective spin-½ particle.
The total charge that has circulated around the loop 118 plays the role of the “momentum” of this particle, such that when these quantities are treated quantum-mechanically and become non-commuting quantum operators, the resulting quantum fluctuations produce tunneling through the central Josephson potential barrier, between the two potential wells. This tunneling plays the role of a Zeeman interaction between the emulated spin and an effective transverse field. This effective transverse field energy can then be tuned by applying a flux ΦX to the DC SQUID 116 loop. The DC SQUID 116 has a total Josephson energy (when it is viewed as a single effective junction 112) that depends on this flux via the Aharonov-Bohm effect, which causes a phase shift between the two parallel Josephson tunneling paths (the two arms of the loop). At ΦX=0, the Josephson energy is maximal, and the tunneling is exponentially small. As ΦX is increased, the two Josephson tunneling amplitudes increasingly cancel each other, lowering the potential barrier and increasing the tunneling exponentially, as shown in FIG. 1C. At ΦX=Φ0/2, the effective Josephson energy goes to zero and the tunneling is maximized. The effective transverse magnetic moment is given approximately by the derivative of this tunneling energy splitting with respect to flux, and as shown in FIG. 1C it goes exponentially to zero as the splitting goes to zero. FIG. 1D illustrates how this tunneling effect can be viewed as a “fluxon” moving out of (or into) the loop; that is, the tunneling is between two states differing by one flux quantum stored in the loop.
From the perspective of spin models, there are two major limitations of this prior art. First, the effective transverse magnetic moment is only large when the barrier height (and therefore the tunneling) is strongly sensitive to flux, which occurs only when the barrier is already low (and thus the tunneling is large) to begin with. In spin Hamiltonian language and shown in FIG. 1C, the transverse moment is only large when the transverse field is itself large. As the transverse field is reduced to zero, the transverse moment decreases exponentially to zero. The result is that the local transverse field terms in the Hamiltonian will be much larger than any implementable transverse interaction terms, a restriction which strongly undercuts the motivation for realizing complex spin models in the first place.
Second, this scheme provides no way to emulate moments corresponding to two independent transverse field directions (i.e. X and Y) in addition to the longitudinal (i.e. Z) moment. The reason for this is simple: the system only provides one accessible magnetic operator—current in the DC SQUID loop 116—to control the coupling between the two eigenstates of Z (spin up and spin down), which we can choose to label X. An effective Y moment would require a second such operator which does not commute with that representing X, since the Pauli X and Y operators are non-commuting. This is an important limitation since many of the quantum applications discussed earlier require both XX and YY interactions, among them holonomic computing, constrained quantum annealing, and most of the many-body spin models of interest (e.g., the Heisenberg model).